Nonintegrability of dynamical systems with homo- and heteroclinic orbits

We consider general n-dimensional systems of differential equations having an (n−2)-dimensional, locally invariant manifold on which there exist equilibria connected by heteroclinic orbits for n≥3. The system may be non-Hamiltonian and have no saddle-centers, and the equilibria are allowed to be the same and connected by a homoclinic orbit. Under additional assumptions, we prove that the monodromy group for the normal variational equation, which is represented by components of the variational equation normal to the locally invariant manifold and defined on a Riemann surface, is diagonalizable or infinitely cyclic if the system is real-meromorphically integrable in the meaning of Bogoyavlenski. We apply our theory to a three-dimensional volume-preserving system describing the streamline of a steady incompressible flow with two parameters, and show that it is real-meromorphically nonintegrable for almost all values of the two parameters.